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Convergent discretisation schemes for transition path theory for diffusion processesJul 12 2019In the analysis of metastable diffusion processes, Transition Path Theory (TPT) provides a way to quantify the probability of observing a given transition between two disjoint metastable subsets of state space. However, many TPT-based methods for diffusion ... More

Uniformly accurate methods for three dimensional Vlasov equations under strong magnetic field with varying directionJul 10 2019In this paper, we consider the three dimensional Vlasov equation with an inhomogeneous, varying direction, strong magnetic field. Whenever the magnetic field has constant intensity, the oscillations generated by the stiff term are periodic. The homogenized ... More

A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS $L^{2}$ spacesApr 09 2019We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as "the" Kaczmarz algorithm), admits important applications in such infinite-dimensional, and non-commutative, settings as are central to spectral ... More

Uniqueness and Stability for the Shock Reflection-Diffraction Problem for Potential FlowMar 29 2019Apr 30 2019When a plane shock hits a two-dimensional wedge head on, it experiences a reflection-diffraction process, and then a self-similar reflected shock moves outward as the original shock moves forward in time. The experimental, computational, and asymptotic ... More

Uniqueness and Stability for the Shock Reflection-Diffraction Problem for Potential FlowMar 29 2019When a plane shock hits a two-dimensional wedge head on, it experiences a reflection-diffraction process, and then a self-similar reflected shock moves outward as the original shock moves forward in time. The experimental, computational, and asymptotic ... More

Line Integral solution of Hamiltonian PDEsMar 15 2019In this paper, we report about recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs), by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we ... More

Superposition, reduction of multivariable problems, and approximationFeb 07 2019We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit reductions schemes ... More

Superposition, reduction of multivariable problems, and approximationFeb 07 2019Mar 06 2019We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit reductions schemes ... More

Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary ProblemsJan 17 2019We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging onto a solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two steady configurations: the ... More

Decomposition of Gaussian processes, and factorization of positive definite kernelsDec 28 2018We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization ... More

Convergence Rates of Gaussian ODE FiltersJul 25 2018Jun 30 2019A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution $x$ and its first $q$ derivatives ... More

Convergence Rates of Gaussian ODE FiltersJul 25 2018A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution $x$ and its first $q$ derivatives ... More

On reproducing kernels, and analysis of measuresJul 11 2018Jul 17 2018Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive ... More

On reproducing kernels, and analysis of measuresJul 11 2018Feb 23 2019Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive ... More

A New Parareal Algorithm for Problems with Discontinuous SourcesMar 14 2018The Parareal algorithm allows to solve evolution problems exploiting parallelization in time. Its convergence and stability have been proved under the assumption of regular (smooth) inputs. We present and analyze here a new Parareal algorithm for ordinary ... More

Convexity of Self-Similar Transonic Shocks and Free Boundaries for the Euler Equations for Potential FlowMar 06 2018Oct 25 2018We are concerned with geometric properties of transonic shocks as free boundaries in two-dimensional self-similar coordinates for compressible fluid flows, which are not only important for the understanding of geometric structure and stability of fluid ... More

A spectral method for nonlocal diffusion operators on the sphereJan 15 2018Jun 08 2018We present algorithms for solving spatially nonlocal diffusion models on the unit sphere with spectral accuracy in space. Our algorithms are based on the diagonalizability of nonlocal diffusion operators in the basis of spherical harmonics, the computation ... More

Realizations and Factorizations of Positive Definite KernelsNov 09 2017Oct 29 2018Given a fixed sigma-finite measure space $\left(X,\mathscr{B},\nu\right)$, we shall study an associated family of positive definite kernels $K$. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic ... More

Sampling with positive definite kernels and an associated dichotomyAug 20 2017We study classes of reproducing kernels $K$ on general domains; these are kernels which arise commonly in machine learning models; models based on certain families of reproducing kernel Hilbert spaces. They are the positive definite kernels $K$ with the ... More

Reproducing kernels and choices of associated feature spaces, in the form of $L^{2}$-spacesJul 26 2017Motivated by applications to the study of stochastic processes, we introduce a new analysis of positive definite kernels $K$, their reproducing kernel Hilbert spaces (RKHS), and an associated family of feature spaces that may be chosen in the form $L^{2}\left(\mu\right)$; ... More

Metric duality between positive definite kernels and boundary processesJun 29 2017We study representations of positive definite kernels $K$ in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes. Our initial results are stated for the most general ... More

Loss of Regularity of Solutions of the Lighthill Problem for Shock Diffraction for Potential FlowMay 19 2017May 07 2018We are concerned with the regularity of solutions of the Lighthill problem for shock diffraction by a convex corned wedge, which can be formulated as a free boundary problem. In this paper, we prove that there is no regular solution that is subsonic up ... More

Positive definite kernels and boundary spacesNov 13 2016We consider a kernel based harmonic analysis of "boundary," and boundary representations. Our setting is general: certain classes of positive definite kernels. Our theorems extend (and are motivated by) results and notions from classical harmonic analysis ... More

Implicit-Explicit difference schemes for nonlinear fractional differential equations with non-smooth solutionsAug 02 2016We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order $0<\beta<1$. From the known structure of the non-smooth solution and by introducing corresponding correction terms, ... More

On the arbitrarily long-term stability of conservative methodsJul 21 2016We show the arbitrarily long-term stability of conservative methods for autonomous ODEs. Given a system of autonomous ODEs with conserved quantities, if the preimage of the conserved quantities possesses a bounded locally finite neighborhood, then the ... More

On the arbitrarily long-term stability of conservative methodsJul 21 2016Oct 11 2018We show the arbitrarily long-term stability of conservative methods for autonomous ODEs. Given a system of autonomous ODEs with conserved quantities, if the preimage of the conserved quantities possesses a bounded locally nite neighborhood, then the global ... More

Explicit high-order symplectic integrators for charged particles in general electromagnetic fieldsMay 04 2016This article considers non-relativistic charged particle dynamics in both static and non-static electromagnetic fields, which are governed by nonseparable, possibly time-dependent Hamiltonians. For the first time, explicit symplectic integrators of arbitrary ... More

Spectral deferred corrections with fast-wave slow-wave splittingFeb 04 2016Jun 13 2016The paper investigates a variant of semi-implicit spectral deferred corrections (SISDC) in which the stiff, fast dynamics correspond to fast propagating waves ("fast-wave slow-wave problem"). We show that for a scalar test problem with two imaginary eigenvalues ... More

Nonuniform sampling, reproducing kernels, and the associated Hilbert spacesJan 27 2016In a general context of positive definite kernels $k$, we develop tools and algorithms for sampling in reproducing kernel Hilbert space $\mathscr{H}$ (RKHS). With reference to these RKHSs, our results allow inference from samples; more precisely, reconstruction ... More

Piecewise-Deterministic Optimal Path PlanningDec 29 2015We consider piecewise-deterministic optimal control problems in which the environment randomly switches among several deterministic modes, and the goal is to optimize the expected cost up to the termination while taking the likelihood of future mode-switches ... More

Peierls substitution and magnetic pseudo-differential calculusJul 22 2015We revisit the celebrated Peierls-Onsager substitution employing the magnetic pseudo-differential calculus for weak magnetic fields with no spatial decay conditions, when the non-magnetic symbols have a certain spatial periodicity. We show in great generality ... More

A uniformly accurate (UA) multiscale time integrator Fourier pseoduspectral method for the Klein-Gordon-Schrodinger equations in the nonrelativistic limit regimeMay 01 2015A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and analyzed for solving the Klein-Gordon-Schr\"{o}dinger (KGS) equations in the nonrelativistic limit regime with a dimensionless parameter $0<\varepsilon\le1$ which is inversely ... More

Induced representations arising from a character with finite orbit in a semidirect productApr 20 2015Aug 11 2015Making use of a unified approach to certain classes of induced representations, we establish here a number of detailed spectral theoretic decomposition results. They apply to specific problems from non-commutative harmonic analysis, ergodic theory, and ... More

Estimating Global Errors in Time SteppingMar 17 2015Apr 13 2016This study introduces new time-stepping strategies with built-in global error estimators. The new methods propagate the defect along with the numerical solution much like solving for the correction or Zadunaisky's procedure; however, the proposed approach ... More

Estimating Global Errors in Time SteppingMar 17 2015May 10 2017This study introduces new time-stepping strategies with built-in global error estimators. The new methods propagate the defect along with the numerical solution much like solving for the correction or Zadunaisky's procedure; however, the proposed approach ... More

Infinite weighted graphs with bounded resistance metricFeb 09 2015Feb 24 2015We consider infinite weighted graphs $G$, i.e., sets of vertices $V$, and edges $E$ assumed countable infinite. An assignment of weights is a positive symmetric function $c$ on $E$ (the edge-set), conductance. From this, one naturally defines a reversible ... More

Generalized Gramians: Creating frame vectors in maximal subspacesJan 28 2015A frame is a system of vectors $S$ in Hilbert space $\mathscr{H}$ with properties which allow one to write algorithms for the two operations, analysis and synthesis, relative to $S$, for all vectors in $\mathscr{H}$; expressed in norm-convergent series. ... More

Quantized linear systems on integer lattices: a frequency-based approachJan 17 2015The roundoff errors in computer simulations of continuous dynamical systems, caused by finiteness of machine arithmetic, can lead to qualitative discrepancies between phase portraits of the resulting spatially discretized systems and the original systems. ... More

Discrete reproducing kernel Hilbert spaces: Sampling and distribution of Dirac-massesJan 10 2015We study reproducing kernels, and associated reproducing kernel Hilbert spaces (RKHSs) $\mathscr{H}$ over infinite, discrete and countable sets $V$. In this setting we analyze in detail the distributions of the corresponding Dirac point-masses of $V$. ... More

Unbounded operators, Lie algebras, and local representationsJun 26 2014We prove a number of results on integrability and extendability of Lie algebras of unbounded skew-symmetric operators with common dense domain in Hilbert space. By integrability for a Lie algebra $\mathfrak{g}$, we mean that there is an associated unitary ... More

Infinite networks and variation of conductance functions in discrete LaplaciansApr 18 2014Mar 08 2015For a given infinite connected graph $G=(V,E)$ and an arbitrary but fixed conductance function $c$, we study an associated graph Laplacian $\Delta_{c}$; it is a generalized difference operator where the differences are measured across the edges $E$ in ... More

Frames and Factorization of Graph LaplaciansApr 05 2014Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $\mathscr{H}_{E}$ of a prescribed infinite (or finite) network. Outside ... More

First integrals for nonlinear dispersive equationsNov 04 2013Given a solution of a semilinear dispersive partial differential equation with a real analytic nonlinearity, we relate its Cauchy data at two different times by nonlinear representation formulas in terms of convergent series. These series are constructed ... More

Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1DOct 15 2013We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law ... More

Uniformly accurate multiscale time integrators for highly oscillatory second order differential equationsDec 20 2012Feb 14 2014In this paper, two multiscale time integrators (MTIs), motivated from two types of multiscale decomposition by either frequency or frequency and amplitude, are proposed and analyzed for solving highly oscillatory second order differential equations with ... More

A Kadec-Pelczyński dichotomy-type theorem for preduals of JBW*-algebrasNov 21 2012We prove a Kadec-Pelczy\'nski dichotomy-type theorem for bounded sequences in the predual of a JBW*-algebra, showing that for each bounded sequence $(\phi_n)$ in the predual of a JBW$^*$-algebra $M$, there exist a subsequence $(\phi_{\tau(n)})$, and a ... More

Multiadaptive Galerkin Methods for ODEs III: A Priori Error EstimatesMay 14 2012The multiadaptive continuous/discontinuous Galerkin methods mcG(q) and mdG(q) for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with ... More

Multi-Adaptive Galerkin Methods for ODEs II: Implementation and ApplicationsMay 12 2012Continuing the discussion of the multi-adaptive Galerkin methods mcG(q) and mdG(q) presented in [A. Logg, SIAM J. Sci. Comput., 24 (2003), pp. 1879-1902], we present adaptive algorithms for global error control, iterative solution methods for the discrete ... More

Explicit Time-Stepping for Stiff ODEsMay 12 2012We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much larger than what ... More

Multi-Adaptive Galerkin Methods for ODEs IMay 12 2012We present multi-adaptive versions of the standard continuous and discontinuous Galerkin methods for ODEs. Taking adaptivity one step further, we allow for individual time-steps, order and quadrature, so that in particular each individual component has ... More

Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave SystemFeb 04 2012Oct 17 2013We are concerned with rigorous mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the nonlinear wave system. This shock diffraction problem can be formulated as a boundary value ... More

Prandtl-Meyer Reflection for Supersonic Flow past a Solid RampDec 31 2011We present our recent results on the Prandtl-Meyer reflection for supersonic potential flow past a solid ramp. When a steady supersonic flow passes a solid ramp, there are two possible configurations: the weak shock solution and the strong shock solution. ... More

Algebraic & definable closure in free groupsAug 29 2011May 14 2012We study algebraic closure and its relation with definable closure in free groups and more generally in torsion-free hyperbolic groups. Given a torsion-free hyperbolic group G and a nonabelian subgroup A of G, we describe G as a constructible group from ... More

Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equationsAug 16 2011Dec 06 2011Purely dispersive partial differential equations as the Korteweg-de Vries equation, the nonlinear Schr\"odinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region ... More

Multigrid methods for two-player zero-sum stochastic gamesJul 08 2011Nov 22 2011We present a fast numerical algorithm for large scale zero-sum stochastic games with perfect information, which combines policy iteration and algebraic multigrid methods. This algorithm can be applied either to a true finite state space zero-sum two player ... More

Tropical Mathematics, Idempotent Analysis, Classical Mechanics and GeometryMay 07 2010A brief introduction to tropical and idempotent mathematics (with an emphasys on idempotent functional analysis) is presented. Applications to classical mechanics and geometry are especially examined.

Shock Reflection-Diffraction Phenomena and Multidimensional Conservation LawsJun 06 2009When a plane shock hits a wedge head on, it experiences a reflection-diffraction process, and then a self-similar reflected shock moves outward as the original shock moves forward in time. The complexity of reflection-diffraction configurations was first ... More

A Parameter-Uniform Finite Difference Method for Multiscale Singularly Perturbed Linear Dynamical SystemsMar 10 2009A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and ... More

First integrals for non linear hyperbolic equationsApr 20 2007Feb 27 2008Given a solution of a nonlinear wave equation on the flat space-time (with a real analytic nonlinearity), we relate its Cauchy data at two different times by nonlinear representation formulas in terms of asymptotic series. We first show how to construct ... More

Classical phase space singularities and quantizationOct 20 2006Simple classical mechanical systems and solution spaces of classical field theories involve singularities. In certain situations these singularities can be understood in terms of stratified Kaehler spaces. We give an overview of a research program whose ... More

Singular Poisson-Kaehler geometry of certain adjoint quotientsOct 20 2006The Kaehler quotient of a complex reductive Lie group relative to the conjugation action carries a complex algebraic stratified Kaehler structure which reflects the geometry of the group. For the group SL(n,C), we interpret the resulting singular Poisson-Kaehler ... More

Nonholonomic Clifford Structures and Noncommutative Riemann--Finsler GeometryAug 09 2004We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, ... More

The Daugavet property of $C^*$-algebras, $JB^*$-triples, and of their isometric predualsJul 13 2004Dec 13 2004A Banach space $X$ is said to have the Daugavet property if every rank-one operator $T:X\longrightarrow X$ satisfies $\|Id + T\| = 1 + \|T\|$. We give geometric characterizations of this property in the settings of $C^*$-algebras, $JB^*$-triples and their ... More

A numerical method for solution of ordinary differential equations of fractional orderFeb 26 2002In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of differential equation ... More

Kaehler spaces, nilpotent orbits, and singular reductionApr 23 2001Aug 13 2002For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata ... More